where \(S_{\rm \nu _0}\)  and \(\tau _{\rm \nu _0}\)  are the flux density and optical depth at the reference frequency \(\nu _0 = 151\)  MHz, respectively, and \(\alpha _{\rm FFA}\)  is the optically thin spectral index. We find the free-free absorption model with radiative cooling to be an excellent fit to the data (see Fig. REF ), with optical depths at 151 MHz and optically thin spectral indices of \(\tau _{\rm FFA}^{A} = 0.22\pm 0.01\)  and \(\alpha _{\rm FFA}^{A } = -0.52\pm 0.01\)  for hotspot A, and \(\tau _{\rm FFA}^{D} = 0.29\pm 0.01\)  and \(\alpha _{\rm FFA}^{D} = -0.50\pm 0.01\)  for hotspot D, respectively. These optical depths would require particle densities of about \(n_e = 3\)  cm\(^{-3}\)  for absorbing clouds with electron temperatures of 8000 K and sizes of 1 kpc, which is around an order of magnitude higher than has been observed for the warm intergalactic medium of our own Galaxy (e.g. [1]}). Furthermore, from Fig. REF  we see that only the hotspots show inverted radio spectra at frequencies above 100 MHz, which suggests that the absorbing material would have to be localized to those regions only, which seems unlikely if the absorbing material was within our Galaxy.
